A polar curve becomes parametric by writing x = r cos θ and y = r sin θ, with θ as the parameter.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The slope
dy/dx is (dy/dθ) ÷ (dx/dθ), exactly as for any parametric curve.
The Product Rule appears
Since r depends on θ, differentiating r cos θ needs the Product Rule. That is where most of the work is.
dr/dθ is not the slope
dr/dθ says how fast the curve leaves the origin. It is not the slope of the tangent line, and confusing the two is the classic error.
Horizontal and vertical tangents
A horizontal tangent needs dy/dθ zero with dx/dθ non-zero. A vertical one is the reverse.
At the origin
Where r = 0, the tangent line to the curve is the line θ equals that angle. That is a useful shortcut on roses.
Convert, then differentiate
Write x = r cos θ and y = r sin θ with r as a function of θ, then use dy/dx = (dy/dθ)/(dx/dθ). The tangent slope in polar coordinates is not dr/dθ, which is a frequent misconception.
dr/dθ is a different quantity
dr/dθ measures how fast the distance from the origin changes as the angle changes. It is a radial rate, not a slope, and it can be zero where the tangent line is far from horizontal.
Horizontal and vertical tangents
Horizontal where dy/dθ is zero and dx/dθ is not; vertical where the reverse holds. Both being zero requires closer examination, since the point may be a cusp.
At the origin, θ gives the tangent
Where a polar curve passes through the origin, the tangent line there is the line at that value of θ. This shortcut identifies the tangents to the petals of a rose without any differentiation.
Step 2: Try It Yourself
Tap and try it out.
- Angle45° = π/4 rad
- x-coordinate0.707
- y-coordinate0.707
- cos 45°0.707
- sin 45°0.707
The cosine is the x-coordinate and the sine is the y-coordinate. They are not merely equal — they are the same two numbers, which is why angles past 90° cause no trouble.
Step 3: Watch an Example
One step at a time.
Watch Kofi Set Up a Polar Slope
Kofi needs dy/dx for the polar curve r = 2.
- Step 1
He writes x = 2 cos θ and y = 2 sin θ, which is a circle of radius 2.
Step 4: Your Turn
Practice makes it stick.
The Conversion
Problem 1 of 2
For r = 2 at θ = 0, what is the x-coordinate?
The Confusion
Problem 2 of 2
Is dr/dθ the slope of the tangent line? 1 yes, 0 no.
Tangents in Polar
1 of 8
For r = 3 at θ = 0, what is the x-coordinate?
2 of 8
For r = 3 at θ = 0, what is the y-coordinate?
3 of 8
dy/dθ = 6 and dx/dθ = 2. What is dy/dx?
4 of 8
dy/dθ = 0 and dx/dθ = 4. Horizontal or vertical tangent? 1 horizontal, 2 vertical.
5 of 8
dy/dθ = 5 and dx/dθ = 0. Horizontal or vertical? 1 or 2?
6 of 8
Which rule is needed to differentiate r cos θ when r depends on θ? 1 product rule, 2 power rule.
7 of 8
Match each quantity with what it measures.
Tap a card on the left to start.
8 of 8
dy/dθ = 12 and dx/dθ = 3. What is dy/dx?
Step 5: Quick Check
Show what you know.
Question 1 of 2
dy/dθ = 15 and dx/dθ = 5. What is dy/dx?
Question 2 of 2
What does dr/dθ measure?
What You Learned
- A polar curve becomes parametric through x = r cos θ and y = r sin θ.
- dy/dx is (dy/dθ) ÷ (dx/dθ), and the Product Rule appears because r varies.
- dr/dθ is not the slope of the tangent.